Scilab Function
Last update : 16/12/2004

svd - singular value decomposition

### Calling Sequence

  `s=svd(X)`
  `[U,S,V]=svd(X)`
  `[U,S,V]=svd(X,0) (obsolete)`
  `[U,S,V]=svd(X,"e")`
  `[U,S,V,rk]=svd(X [,tol])`

### Parameters

- `X`: a real or complex matrix
- `s`: real vector (singular values)
- `S`: real diagonal matrix (singular values)
- `U,V`: orthogonal or unitary square matrices (singular vectors).
- `tol`: real number

### Description

`[U,S,V] = svd(X)` produces a diagonal matrix `S` , of the same dimension as `X` and with nonnegative diagonal elements in decreasing order, and unitary matrices `U` and `V` so that `X = U*S*V'`.

`[U,S,V] = svd(X,0)` produces the "economy size" decomposition. If `X` is m-by-n with m > n, then only the first n columns of `U` are computed and `S` is n-by-n.

`s = svd(X)` by itself, returns a vector `s` containing the singular values.

`[U,S,V,rk]=svd(X,tol)` gives in addition `rk`, the numerical rank of `X` i.e. the number of singular values larger than `tol`.

The default value of `tol` is the same as in `rank`.

### Examples

    X=rand(4,2)*rand(2,4)
    svd(X)
    sqrt(spec(X*X'))

### See Also

`rank`, `qr`, `colcomp`, `rowcomp`, `sva`, `spec`,

### Used Function

svd decompositions are based on the Lapack routines DGESVD for real matrices and ZGESVD for the complex case.